Properties

Label 1.625.abw
Base Field $\F_{5^{4}}$
Dimension $1$
Ordinary Yes
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{5^{4}}$
Dimension:  $1$
L-polynomial:  $1 - 48 x + 625 x^{2}$
Frobenius angles:  $\pm0.0903344706017$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-1}) \)
Galois group:  $C_2$

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $1$
Slopes:  $[0, 1]$

Point counts

This isogeny class contains a Jacobian, and hence is principally polarizable.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 578 389572 244120034 152587560960 95367428686658 59604644839641412 37252902989549408354 23283064365583459000320 14551915228373202202575938 9094947017729464388086726852

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 578 389572 244120034 152587560960 95367428686658 59604644839641412 37252902989549408354 23283064365583459000320 14551915228373202202575938 9094947017729464388086726852

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5^{4}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-1}) \).
All geometric endomorphisms are defined over $\F_{5^{4}}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
1.625.bw$2$(not in LMFDB)
1.625.ao$4$(not in LMFDB)
1.625.o$4$(not in LMFDB)